SINR for DS-CDMA with random spreading

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SINR for DS-CDMA with random spreading

P.-H. Chiang, D.-B. Lin and H.-J. Li

Abstract: For wireless multipath fading channels, the performance of a direct-sequence code-division multiple-access (DS-CDMA) system is degraded by the interferences, including the multipath interference (MPI) and the multiuser interference (MUI). For the downlink DS-CDMA with random spreading, we derive the variances of the interferences and obtain the signal-to-interference-plus-noise ratio (SINR). The simple and elegant expressions for the variances of the interferences provide useful insights, while the corresponding SINR representation is helpful in evaluating the bit-error-rate (BER) performance that takes the impact of the interferences into consideration. In order to demonstrate our approach, an example regarding the performance analysis of the two-input–single-output (2ISO) space–time block coded DS-CDMA with RAKE reception is also provided.

1 Introduction

For wireless multipath fading channels, the performance of a direct-sequence code-division multiple-access (DS-CDMA) system is degraded by the interferences, which are classiﬁed into two categories [1]: 1) the multipath interference (MPI), consisting of the intersymbol inter-ference (ISI) and the interpath interinter-ference (IPI), owing to multipath propagation; and 2) the multiuser interference (MUI) from other active users.

Recently, Bjerke et al. analysed the bit-error-rate (BER) performances of the downlink DS-CDMA with various antenna diversity schemes[2], including both transmit and receive diversity. However, they assumed perfect spreading and hence the impacts of the MPI and the MUI on the system performance are not taken into account.

In this paper, for the downlink DS-CDMA with random spreading, we derive the variances of the interferences and obtain the signal-to-interference-plus-noise ratio (SINR). The simple and elegant expressions for the variances of the interferences can provide useful insights. Moreover, the impacts of the interferences can be included in the derived BER expressions by applying the concept of effective signal-to-noise ratio (SNR) to the results in[2]. To demonstrate our strategy, an example regarding the performance analysis of the two-input–single-output (2ISO) space–time block coded DS-CDMA (STBC-DS-CDMA) with RAKE recep-tion is also provided.

The rest of this paper is organised as follows. In Section 2, the system model and the statistical properties are described. An example for demonstrating our strategy is given in Section 3. Then some numerical results are shown in Section 4, whereas conclusions are drawn in Section 5.

2 Preliminaries

In this section, the system model of the DS-CDMA is described and some statistical properties comprising the variances of the interferences are given. To simplify the derivation, we consider the SISO synchronous downlink transmission, as shown in Fig. 1.

2.1 System model

Let Xk,idenote the ith information symbol of the kth user, and TS represent the symbol duration. After short code spreading, the transmitted chip sequence for all K users in the ith symbol interval is expressed as

~si;l¼

XK1 k¼0

Xk;ibk;l; for 0 l L 1 ð1Þ

where {bk,l, 0r l r L1} is the random spreading sequence, taking on the values of 1=pffiffiffiL with equal probability. Thus, TS¼ LTC, where L and TCrepresent the spreading factor (i.e. the processing gain) and the chip duration (i.e. the reciprocal of the system bandwidth), respectively. Assuming that ~si;lis zero for lo 0 and l Z L,

the total transmitted baseband sequence is written as sl¼

X1 i¼1

~si;liL ð2Þ

In this paper, we consider the wide-sense stationary uncorrelated scattering (WSSUS) Rayleigh fading channel which can be modelled as a tapped delay line model[3]with ﬁxed tap spacing TC. Regarding the channel with M taps and assuming perfect synchronisation and power control, the received baseband sequence of the desired user, denoted by the dth user, can be expressed as

rd;l¼ X M1 m¼0 hd;m;lslmþ nd;l ¼ X 1 i¼1 X M1 m¼0 hd;m;l~si;lmiLþ nd;l ð3Þ

where hd,m,l is the tap coefﬁcient with mean zero and variance s2

d;m, and nd,lis the AWGN with mean zero and variance N0. Assuming the channels with respect to different E-mail: [email protected]

P.-H. Chiang and H.-J. Li are with Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, Republic of China D.-B. Lin is with Department of Electronic Engineering, National Taipei University of Technology, Taipei, Taiwan, Republic of China

rThe Institution of Engineering and Technology 2006 IEE Proceedings online no. 20045235

doi:10.1049/ip-com:20045235

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users are identical, the subscript d of hd,m,l, s2d;m, and nd,lis omitted for notation simplicity in the following sections. As shown in Fig. 2, assuming MoL and rearranging rd,l symbol by symbol as rd,i,l¼ rd,iL+lyield

rd;i;l¼ Pl m¼0 hm;i;l~si;lmþ P M1 m¼lþ1 hm;i;l~si1;lmþLþ ni;l; if 0 l M 2; P M1 m¼0

hm;i;l~si;lmþ ni;l; if M 1 l L 1;

8 > > > > > > < > > > > > > : ð4Þ where hm;i;l¼ hm;iLþland ni;l¼ niLþl. Inserting (1) into (4)

gives rd;i;l¼ P K1 k¼0 Xk;iP l m¼0 hm;i;lbk;lmþ P K1 k¼0 Xk;i1 P M1 m¼lþ1 hm;i;lbk;lmþL þni;l;if 0 l M 2; P K1 k¼0 Xk;i P M1 m¼0

hm;i;lbk;lmþ ni;l;if M 1 l L 1 8 > > > > > > > < > > > > > > > : ð5Þ Assume the RAKE receiver has F fingers and Fr M. In addition, for the finger f, 0r f r F1, we define (see Fig. 2)

~rd;f ;i;l¼

rd;i;lþf; if 0 l L f 1;

rd;iþ1;lþf L; if L f l L 1

ð6Þ Then substituting (5) into (6) produces

~rd;f ;i;l¼ P K1 k¼0 Xk;iP lþf m¼0 hm;i;lþfbk;lþf m þP K1 k¼0 Xk;i1 P M1 m¼lþf þ1 hm;i;lþfbk;lþf mþLþni;lþf; if 0 l M f 2; P K1 k¼0 Xk;i P M1 m¼0 hm;i;lþfbk;lþf mþni;lþf; if M f 1 l L f 1; P K1 k¼0 Xk;iþ1 P lþf L m¼0 hm;i;lþf Lbk;lþf mL þP K1 k¼0 Xk;i P M1 m¼lþf Lþ1 hm;i;lþf Lbk;lþf mþni;lþf L; if L f l L 1 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > : ð7Þ Accordingly, at the receiver of the desired user, the code-matched ﬁlter with respect to the f th path performs despreading on ~rd;f ;i;l; 0 l L 1

and outputs signal for the ith symbol as

Rd;f ;i¼

XL1 l¼0

~rd;f ;i;lbd;l

¼ Hf ;iXd;iþ Pd;f ;iþ Sd;f ;iþ Ud;f ;iþ Nf ;i ð8Þ

where, after changing variables, the multiplicative distortion (MD) Hf,i, IPI Pd,f,i, ISI Sd,f,i, MUI Ud,f,i, and noise Nf,iare, respectively, given by Hf ;i¼ 1 L XL1 l¼0 hf ;i;l ð9Þ Pd;f ;i ¼Xd;i X M2 l¼f Xl m¼0 m6¼f hm;i;lbd;lmbd;lf 0 B @ þ X L1 l¼M1 X M1 m¼0 m6¼f hm;i;lbd;lmbd;lf þX f1 l¼0 X M1 m¼lþ1 m6¼f hm;i;lbd;lmþLbd;lf þL 1 C A ð10Þ Sd;f ;i¼Xd;i1 X M2 l¼f X M1 m¼lþ1 hm;i;lbd;lmþLbd;lf þ Xd;iþ1 Xf1 l¼0 Xl m¼0 hm;i;lbd;lmbd;lf þL ð11Þ i i −1 i + 1 1 2 m = 0 i − 1 i i + 1 1 4 m = 1 = + 0 2 4 i m 0 1 3 4 i 4 0 2 3 m = 2 0 = + i 3 4 3 0 2 3 i − 1 _i_{+ 1} i − 1 i i + 1 1 1 2

{

S i,((l −m))L}l = 0 ∼ L−1

{

r∼_d,3,i,l

}

L− 1 l = 0

{

r∼_d,1,i,l

}

L− 1 l = 0

{

r_d,i,l

}

L− 1 l = 0

Fig. 2 Graphical illustration of received signals for M¼ 3, L ¼ 5

despreading modulation spreading L − 1 l = 0 {b_d,l} channel BS to d th MS ML detection T_C R_d,1,i de- modulation

form other users

R_d,F_−1,i X_k,i ˆ X L −1 l = 0 r user k RAKE receiver user d l = 0 L−1 {r_d,i,l} R_d,0,i X_k,i _despreading despreading T_C L − 1 l = 0 {s_i,l} {b_k,l}L − 1 l = 0 L−1 l =0 {r_d,1,i,l} L −1 l = 0 {r_d,F_−1,i,l}

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Ud;f ;i¼ XK1 k¼0 k6¼d Xk;i X M2 l¼f Xl m¼0 hm;i;lbk;lmbd;lf " þ X L1 l¼M1 X M1 m¼0 hm;i;lbk;lmbd;lf þX f1 l¼0 X M1 m¼lþ1 hm;i;lbk;lmþLbd;lf þL ! þ Xk;i1 X M2 l¼f X M1 m¼lþ1 hm;i;lbk;lmþLbd;lf þ Xk;iþ1 X f1 l¼0 Xl m¼0 hm;i;lbk;lmbd;lf þL # ð12Þ Nf ;i¼ XL1 l¼0 ni;lbd; lððfÞÞL ð13Þ

Here ððÞÞL denotes the modulo-L operation, and the

subscript d of Hd,f,iand Nd,f,iis omitted. Then, by changing the order of the summations and rearranging the resultant terms, (10)–(12) can be rewritten as

Pd;f ;i ¼Xd;i X M1 m¼0 m6¼f XL1 l¼m hm;i;lbd;lmbd;lf 0 B @ þX M1 m¼1 m6¼f X m1 l¼0 hm;i;lbd;lmþLbd;lf þL X M1 m¼f þ1 X m1 l¼f hm;i;lbd;lmþLbd;lf X f1 m¼0 Xf1 l¼m hm;i;lbd;lmbd;lf þL ! ð14Þ Sd;f ;i ¼Xd;i1 X M1 m¼f þ1 X m1 l¼f hm;i;lbd;lmþLbd;lf þ Xd;iþ1 X f1 m¼0 X f1 l¼m hm;i;lbd;lmbd;lf þL ð15Þ Ud;f ;i ¼ X K1 k¼0 k6¼d Xk;i X M1 m¼0 XL1 l¼m hm;i;lbk;lmbd;lf " þX M1 m¼1 X m1 l¼0 hm;i;lbk;lmþLbd;lf þL X f1 m¼0 Xf1 l¼m hm;i;lbk;lmbd;lf þL X M1 m¼f þ1 X m1 l¼f hm;i;lbk;lmþLbd;lf ! þ Xk;i1 X M1 m¼f þ1 Xm1 l¼f hm;i;lbk;lmþLbd;lf þ Xk;iþ1 X f1 m¼0 X f1 l¼m hm;i;lbk;lmbd;lf þL # ð16Þ

From (13), for random spreading, it is readily recognised that Nf,iis still the AWGN with mean zero and variance N0. It is assumed that the symbols fXk;i; 8k and ig are

independently and identically distributed (i.i.d.) with mean zero and variance ES (symbol energy) and the tap coefﬁcientsfhm;i;l; 0 m M 1g are mutually

uncorre-lated. Thereupon, even though the distributions of the interferences, including Pd,f,i, Sd,f,i, and Ud,f,iare not easy to identify, the following is readily veriﬁed: 1) they are mutually uncorrelated; 2) they are all uncorrelated to the MD Hf,iand noise Nf,i, as well; 3) they are all zero-mean. Furthermore, since independent Gaussian noise results in the smallest capacity, it is reasonable to model these interferences as Gaussian random variables and hence to achieve the performance bound [4]. Indeed, (8) can be rewritten as

Rd;f ;i¼ Hf ;iXd;iþ Wd;f ;i ð17Þ

where Wf ;i¼ Pd;f ;iþ Sd;f ;iþ Ud;f ;iþ Nf ;i is the equivalent

AWGN with mean zero and variance s2

Wd;f ¼ s2Pd;f þ s 2 Sd;f þ s 2 Ud;f þ N0.

2.2 Statistical properties

In this subsection, the statistical properties regarding the channel tap coefﬁcients, MD, and interferences are clariﬁed.

For WSSUS Rayleigh fading channels with classical Doppler spectrum, the correlation of tap coefﬁcients is given by[3] E hm;i;lhm0_;i0_;l0 h i ¼ s2 mJ0f2pfDTC½ði i0ÞL þ l lð 0Þgdmm0 ð18Þ where s2

m is the fading power of the mth tap, J0( ) is the zeroth-order Bessel function of the ﬁrst kind, fD is the maximum Doppler frequency, and dij is the Kronecker delta. Considering the exponential power delay proﬁle ( PDP) [3]with the constraintPM_m_¼01s2

m¼ 1, we have s2_m¼ð1 lÞ 1 lMl m ð19Þ where l¼ e1=d_{, and the delay control d dominates}

the normalised root-mean-square (RMS) delay spread trms/TC.

2.2.1 Variance of the multiplicative distortion

: In light of (9), it is clear that the MD Hf,ihas mean zero and hence variance s2Hf ¼ E½jHf ;ij 2 ¼s 2 f L2 XL1 l¼0 XL1 l0_¼0 J0½2pfDTCðl l0Þ ¼s 2 f L2 XL1 l¼Lþ1 L jlj ð ÞJ0ð2pfDTClÞ ð20Þ

2.2.2 Variances of the interferences

: From (14)

and (15), utilising the fact that the information symbols, channel tap gains, and random spreading sequences are mutually independent, the variances of the IPI Pd;f ;iand ISI

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Sd;f ;i are calculated as follows. s2_P_d;f ¼ E½jPd;f ;ij2 ¼ES L2 X M1 m¼0 m6¼f L m ð Þ þ m ½ s2m 8 > < > : X M1 m¼f þ1 m f ð Þs2m Xf1 m¼0 f m ð Þs2m ) ¼ES L2 X M1 m¼0 m6¼f L m fj j ð Þs2m ð21Þ s2 Sd;f ¼ E½jSd;f ;ij 2 ¼ES L2 X M1 m¼f þ1 m f ð Þs2mþ X f1 m¼0 f m ð Þs2m " # ¼ES L2 X M1 m¼0 m6¼f m f j js2m ð22Þ

Thereupon, summing (21) and (22) yields the variance of the MPI as s2Pd;f þ s 2 Sd;f ¼ ES L X M1 m¼0 m6¼f s2m ¼ES L 1 s 2 f ES L ð23Þ

Similarly, according to (16), the variance of the MUI Ud,f,iis derived as s2_U_d;f ¼ E½jUd;f ;ij2 ¼ES L2 X K1 k¼0 k6¼d X M1 m¼0 L m fj j þ m fj j ð Þs2m ¼ES L ðK 1Þ ð24Þ

Consequently, from (23) and (24), the power of the total interference can be expressed by a simple formula as

s2_P_d;fþ s2Sd;f þ s 2 Ud;f ¼ ES L ½ð1 s 2 fÞ þ ðK 1Þ ð25Þ

Noteworthily, an upper bound of (25) can be obtained by considering the most dispersive condition of the exponential PDP. In light of (19), for this extreme case, d-N, l-1, and s2f ! 1=M, i.e., the exponential PDP approaches the

uniform PDP. Therefore, under this extreme channel condition, the power of the total interference is given by

s2_P d;f þ s 2 Sd;f þ s 2 Ud;f ¼ ES L 1 1 M þ K 1ð Þ ES L ½1þ K 1ð Þ ¼ ES L K ð26Þ

which shows that the MPI can be treated as the MUI from an additional user. In other words, the total interference can be approximated as the MUI contributed by K users. This conﬁrms the assumption in [1, p. 1138]. Moreover, it is evident that the number of users K dominates the quantity of the total interference whereas the spreading factor L stands for the inherent interference rejection ability of a DS-CDMA system.

In the subsequent sections, the subscript d is omitted for notation simplicity.

3 2ISO DS-CDMA

To demonstrate our strategy, an example for the perfor-mance analysis of the 2ISO STBC-DS-CDMA system in the quasi-static (QS) channel is provided. According to the system model mentioned in Section 2.1, we ﬁrstly review the mechanisms of the STBC-DS-CDMA, introduce its max-imum-likelihood (ML) detection for the receiver design, and then derive the SINR and BER for performance analysis.

3.1 Space–time transmission

For user d and the symbol intervals 2i+0 and 2i+1, a pair of information symbols Xð 2iþ0; X2iþ1Þ is ST block encoded

according to the transmission matrix as[5]

Space ! Time # X2iþ0 X2iþ1 X 2iþ1 X2iþ0 ð27Þ More speciﬁcally, at symbol instant 2i+0, X2i+0and X2i+1 are transmitted from antenna 0 and 1, respectively. Then, at symbol instant 2i+1,X

2iþ1 and X2iþ0 are transmitted

from antenna 0 and 1, respectively.

Let H_{f ;}ðgÞ_2iþ0 and H_{f ;}ðgÞ_2iþ1 be the spatially identical and independent MDs for the gth transmit antenna, g¼ 0,1. Also, assuming the channel is QS over a space–time code-word duration (i.e. 2TS), the piecewise-constant MD for the ith space–time codeword is deﬁned as HðgÞ_{f ;i}_9H_{f ;}ðgÞ_2iþ0 ¼ H_{f ;}ðgÞ_2iþ1with mean zero and variance s2

f. Accordingly, from

(17) and (27), the output for the f th ﬁnger is[2] rf ;i¼ Hf ;ixiþ wf ;i Rf ;2iþ0 R f ;2iþ1 " # ¼ H ð0Þ f ;i H ð1Þ f ;i Hð1Þ_{f ;i} Hð0Þf ;i 2 4 3 5 X2iþ0 X2iþ1 þ Wf ;2iþ0 W f ;2iþ1 " # ; ð28Þ where wf ;i CNð0; s2WfI2Þ and s 2 Wf ¼ 2ðs 2 Pf þ s 2 Sfþ s2

Uf þ N0Þ. The factor 2 is a result of the sharing of the transmit power for two antennas.

3.2 Maximum-likelihood detection

Performing the space–time matched ﬁltering on rf ; igives[6]

r _ f ;i¼ K f ;iHf ;i H rf ;i ¼ Kf ;iHHf ;iHf ;i xiþ Kf ;iHHf ;i wf ;i ¼ H_f ;ixiþ w _ f ;i ð29Þ where Kf ;i¼ a1f ;iI2, H _ f ;i¼ af ;iI2, af ;i¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P1 g¼0jH ðgÞ f ;ij 2 q , r _ f ;i¼ ½R _ f ;2iþ0 R _ f ;2iþ1T, and w _ f ;i¼ ½W _ f ;2iþ0 W _ f ;2iþ1T.

Notice that Kf ;iis chosen such that w _

f ;i CNð0; s2WfI2Þ. Owing to that H_f ;i is diagonal, (29) can be separated into

two equations as R _ f ;2iþt¼ af ;iX2iþtþ W _ f ;2iþt; for t¼ 0; 1 ð30Þ

From (30), stacking upfR_f ;2iþt; 0 f F 1g results in

the signal model for the ML detection as r _ 2iþt¼ aiX2iþtþ w _ 2iþt; for t¼ 0; 1 ð31Þ where r_2iþt¼ ½R _ 0;2iþt R _ 1;2iþt R _

F1;2iþtT, ai¼ ½a0;i a1;i

aF1;iT and w _ 2iþt¼ ½W _ 0;2iþt W _ 1;2iþt W _ F1;2iþtT.

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Since w_2iþt CN 0; RRð Þ, where R ¼ diagfsW0; sW1; . . . ; sWF1g, then the ML detection is given by

^

X2iþt¼ arg min X ð r_2iþt aiXÞ H ðRRÞ1ð r_2iþt aiXÞ ¼ arg min X kS1ð r_2iþt aiXÞk2; for t¼ 0; 1 ð32Þ where R1represents the noise balancing. Furthermore, let

r _ 2iþt¼ S 1__r 2iþt¼ aiX2iþtþ w _ 2iþt; for t¼ 0; 1 ð33Þ with a_i ¼ R1ai and w _ 2iþt¼ S 1_w_ 2iþt. Then, (32) is equivalent to ^

X2iþt¼ arg min X k r_2iþt aiXk2 ¼ arg min X Y2iþt X j j2; for t¼ 0; 1 ð34Þ where Y2iþt¼ a T i r _ 2iþt kai k

2 ¼ X2iþtþ Z2iþt; for t¼ 0; 1 ð35Þ

and Z2iþt ¼ aTi w _

2iþt =k aik

2_{. From (33)–(35), it is clear}

that the 2Fth order diversity channel is transformed into the equivalent AWGN channel via the maximum-ratio combining (MRC).

3.3 SINR and BER

According to (35), the total instantaneous SINR is derived as

g¼ E½jX2iþtj2=E½jZ2iþtj2

¼ kaik2Es¼ kR1aik2Es ¼X F1 f¼0 a2 f ;iEs s2 Wf ¼X 1 g¼0 XF1 f¼0 jHðgÞ_{f ;i}j2Es s2 Wf ¼X 1 g¼0 XF1 f¼0 rðgÞ_f ð36Þ

where rðgÞ_f is the instantaneous SINR of theðg; f Þth branch. Also, g is of probability density function ( pdf) as [2, eq. (18)]

prð Þ ¼x XF1 f¼0 Bf1 Gf þBf2 G2_f x ! e x Gf_; _x₀ _ð37Þ where Bf1 ¼ 2A2f PF1 m¼0 m6¼f Gm=ðGm GfÞ, Bf2¼ A 2 f, Af ¼ QF1 m¼0 m6¼f Gf=ðGf GmÞ and Gf ¼ E½g ðgÞ f .

From (34), for BPSK modulation, ^X2iþt is of conditional

BER Qðpffiffiffiffiffi2gÞ and hence its average BER is derived as [2, eq. (19)] Pb¼ Z 1 0 prðxÞQð ffiffiffiffiffi 2x p Þdx ¼X F1 f¼0 Bf1 2 ð1 mfÞ þ Bf2 4 ð2 3mfþ m 3 fÞ ð38Þ where mf ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gf=ðGfþ 1Þ p

Based on this example regarding the 2ISO DS-CDMA system, by applying the concept of effective SNR to the results in[2], one can theoretically investigate the impact of the MPI and MUI on performances of DS-CDMA systems with various antenna diversity schemes.

4 Numerical results

For all numerical results, the parameters are detailed as follows. First, the system bandwidth is 800 kHz, thus the chip duration is TC¼ 1.25 ms; secondly, for exponential PDP, the number of uncorrelated paths and the delay control are M¼ 6 and d ¼ 2, respectively, which correspond to the RMS delay spread trms¼ 1.74 ms; thirdly, the channel is QS; fourthly, the modulation is BPSK; ﬁnally, the number of ﬁngers is F¼ 1, 2, y, 6. In addition to the BERs of the 2ISO DS-CDMA systems, the BERs of the SISO systems are also provided as benchmarks.

Figure 3 shows the BERs of various DS-CDMA systems with K¼ 10 and L ¼ 127. It is evident that the interferences, comprising the MPI and MUI, have great impact on system performance such that irreducible error ﬂoors occur. In addition, it is clear that the 2ISO systems beneﬁt from larger diversity gain and hence perform better than the SISO systems. Thereupon, combining the path and antenna diversities does enhance the signal component and provide better interference reduction.

Figure 4 illustrates the error ﬂoors corresponding to different spreading factors (i.e. L¼ 15, 31, 63, 127, 255, 511, 1023, and 2047), provided that K¼ 10. It is conﬁrmed that the larger the spreading factor, the greater the interference rejection ability of DS-CDMA systems.

0 5 10 15 20 25 30 35 40 10−5 10−4 10−3 10−2 10−1 100 E_S/ N₀, dB BER SISO 2ISO K = 10 L = 127 F = 1,2,...,6

Fig. 3 BER against SNR

102 103 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 E_S/N₀ = 20dB BER SISO 2ISO F = 1,2,...,6 K = 10 spreading factor

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Figure 5 depicts the error ﬂoors with respect to different numbers of users, given that L¼ 127. For F ¼ 6, if the permissible BER is 103, then the permissible numbers of users for the SISO and 2ISO systems are about 12 and 18, respectively. Consequently, by installing one additional transmit antenna at the base station, the system capacity is increased about 6 users.

5 Conclusions

In this paper, for the downlink DS-CDMA with random spreading, we derive variances of the interferences, including

the MPI and MUI, and obtain the SINR. The simple and elegant expressions for variances of the interferences provide useful insights, while the corresponding SINR representa-tion is helpful in evaluating the BER performance that takes the inﬂuence of the interferences into account. An example regarding performance analysis of the 2ISO STBC-DS-CDMA with RAKE reception is also provided. Based on our approach, by applying the concept of effective SNR to the results in[2], one can theoretically investigate the impact of the interferences on performances of DS-CDMA systems with various antenna diversity schemes.

6 Acknowledgments

This work was supported by the National Science Council, Republic of China, under Grant NSC 92-2219-E-002-010 and the Ministry of Education Program for promoting academic excellence of universities under Grant 89E-FA06-2-4-7.

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two-branch transmit diversity block coded OFDM systems in time-varying multipath Rayleigh fading channels’, IEEE Trans. Veh. Technol., 2005, 54, pp. 136–148 10−7 2 4 6 8 10 12 14 16 18 20 10−6 10−5 10−4 10−3 10−2 10−1 100 E_S/ N₀ = 20dB BER SISO 2ISO F = 1,2,...,6 L = 127 number of users